Probability question

[henryzz]

I though of an interesting question when playing freecell on freecell.net yesterday. I have played 116 games of standard freecell and have won 83.62% of them. My longest streak is 17 games. What I wanted to find out was given a random distribution what is the longest streak I should have expected in those games and the probability of having each length of streak.

I think the probability of winning 17 consecutive games is 83.62%^17 = 4.78%. There are 100 possible blocks of 17 in 116(1-17, 2-18 etc) so should I expect 4.78 streaks of 17? Am I correct?
What about my other questions on what is the longest expected streak and the probability of having each length of streak?

[wblipp]

There are two issues that you have ignored. They are usually ignored for informal purposes such as this, but deserve being mentioned before we ignore them.

1. Is your winning probability constant, or are you improving over time? Or even fluctuating more randomly based on how tired you are?

2. Even if it is fixed, how well does this finite sample estimate your winning probability?

If we agree to ignore these issues, your results are correct for the expected number of streaks of 17, although it counts a streak of 18 as two streaks of 17. This works because the Expected Value of A+B is the Expected Value of A plus the Expected Value of B, even when A and B are dependent events. In this context, "A" would be the probability of winning games 1-17 and B would be the probability of winning 2-18.

Your other questions are harder because they do not share this simplifying calculation method. I would tackle the expected longest streak by calculations on a Markov Chain with state space of (i,j) with i=current run and j=longest run so far. If nobody comes up with a better method and you are still interested, I'll provide more details about how to do that.

[henryzz]

Originally Posted by wblipp

There are two issues that you have ignored. They are usually ignored for informal purposes such as this, but deserve being mentioned before we ignore them.

1. Is your winning probability constant, or are you improving over time? Or even fluctuating more randomly based on how tired you are?

2. Even if it is fixed, how well does this finite sample estimate your winning probability?

If we agree to ignore these issues, your results are correct for the expected number of streaks of 17, although it counts a streak of 18 as two streaks of 17. This works because the Expected Value of A+B is the Expected Value of A plus the Expected Value of B, even when A and B are dependent events. In this context, "A" would be the probability of winning games 1-17 and B would be the probability of winning 2-18.

Your other questions are harder because they do not share this simplifying calculation method. I would tackle the expected longest streak by calculations on a Markov Chain with state space of (i,j) with i=current run and j=longest run so far. If nobody comes up with a better method and you are still interested, I'll provide more details about how to do that.

1. Improving, tiredness, some games are harder than others. All of these things change the probability. For example in the last 20 games I have lost only once which is 95% won(and increased my top streak to 18 before losing)

2. Not brilliantly. Could do with more precision in the probability. Plus I am improving currently(now at 84.13% after 126 games) The probability changed from 84.00 to 84.13 for just one win. That changes the probability of a streak of 17 from 5.16 to 5.30. These are still big changes for one game(just look at those values in comparison with the original 4.78%).

It's probably not worth spending much more effort on this as the accuracy of any results will not be very high.

[davieddy]

Originally Posted by wblipp

There are two issues that you have ignored. They are usually ignored for informal purposes such as this, but deserve being mentioned before we ignore them.

1. Is your winning probability constant, or are you improving over time? Or even fluctuating more randomly based on how tired you are?

2. Even if it is fixed, how well does this finite sample estimate your winning probability?

If we agree to ignore these issues, your results are correct for the expected number of streaks of 17, although it counts a streak of 18 as two streaks of 17. This works because the Expected Value of A+B is the Expected Value of A plus the Expected Value of B, even when A and B are dependent events. In this context, "A" would be the probability of winning games 1-17 and B would be the probability of winning 2-18.

Your other questions are harder because they do not share this simplifying calculation method. I would tackle the expected longest streak by calculations on a Markov Chain with state space of (i,j) with i=current run and j=longest run so far. If nobody comes up with a better method and you are still interested, I'll provide more details about how to do that.

My best winning steak was >200.
Eventually I got too bored/drunk to avoid a loss, and admitted defeat.

There are impossible starting positions, and similar ...king difficult
ones.

Re probability Henryzz, revisit the coin tossing thread.

David